Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave patterns.

Figure 1

(Color online) The possible spatiotemporal patterns in coupled logistic lattice (1, 5) with $L=8$ for different initial conditions. (a) Single standing wave, (b) double standing waves, (c) single standing wave with a temporal period-2 orbit in one half and period-4 orbit in the other half of the lattice and, (d) synchronized chaos.

Figure 2

The phase diagram in the $k_1\text{−}{k}_2$ plane for the modified map (6) with $f\left(x\right)=\mu x(1-x)$, $\mu =3.5732$, showing the regions which correspond to the periodic solution of period 2 (dark region).

Figure 3

(Color online) Standing wave patterns obtained from Eq. (7) for $L=8$: (a) standing wave with period-1 UPO at nodes and period-2 UPO at antinodes and (b) period-2 UPO at nodes and period-4 UPO at antinodes.